Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 6C / a / Solution

Probability enters state $n$ through a birth from $n-1$ or a pair-death event from $n+2$, and leaves through either event at state $n$. Thus the Kramers-Moyal master equation iswith probabilities and impossible transition rates taken as zero outside the nonnegative states.

A birth changes $n$ by $+1$, while a death event changes it by $-2$. Applying these increments to the first moment givesConsequently every steady state must satisfy